Properties

Label 304.7.e.f.113.6
Level $304$
Weight $7$
Character 304.113
Analytic conductor $69.936$
Analytic rank $0$
Dimension $30$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [304,7,Mod(113,304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(304, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("304.113");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 304.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(69.9364414204\)
Analytic rank: \(0\)
Dimension: \(30\)
Twist minimal: no (minimal twist has level 152)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 113.6
Character \(\chi\) \(=\) 304.113
Dual form 304.7.e.f.113.25

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-33.4715i q^{3} +71.5877 q^{5} -51.8109 q^{7} -391.339 q^{9} +O(q^{10})\) \(q-33.4715i q^{3} +71.5877 q^{5} -51.8109 q^{7} -391.339 q^{9} -510.930 q^{11} +2026.71i q^{13} -2396.14i q^{15} +8792.00 q^{17} +(6072.98 - 3188.23i) q^{19} +1734.19i q^{21} -22866.7 q^{23} -10500.2 q^{25} -11302.0i q^{27} -41913.8i q^{29} -49006.5i q^{31} +17101.6i q^{33} -3709.02 q^{35} +6756.83i q^{37} +67836.9 q^{39} +39497.7i q^{41} +36231.6 q^{43} -28015.0 q^{45} +142082. q^{47} -114965. q^{49} -294281. i q^{51} -1332.46i q^{53} -36576.3 q^{55} +(-106715. - 203271. i) q^{57} -90431.3i q^{59} +111587. q^{61} +20275.6 q^{63} +145087. i q^{65} +152853. i q^{67} +765382. i q^{69} -572540. i q^{71} -231113. q^{73} +351457. i q^{75} +26471.7 q^{77} -421938. i q^{79} -663581. q^{81} -767322. q^{83} +629399. q^{85} -1.40292e6 q^{87} -486637. i q^{89} -105005. i q^{91} -1.64032e6 q^{93} +(434750. - 228238. i) q^{95} +1.34102e6i q^{97} +199947. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 30 q + 720 q^{7} - 8670 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 30 q + 720 q^{7} - 8670 q^{9} + 2524 q^{11} + 9700 q^{17} - 4014 q^{19} + 39376 q^{23} + 110742 q^{25} + 19976 q^{35} + 266500 q^{39} + 106788 q^{43} - 91360 q^{45} - 222756 q^{47} + 593586 q^{49} - 540936 q^{55} - 545972 q^{57} - 242640 q^{61} + 377716 q^{63} + 545964 q^{73} - 272356 q^{77} + 2189926 q^{81} - 1542652 q^{83} - 826908 q^{85} + 2729572 q^{87} - 2139912 q^{93} - 2142716 q^{95} + 293012 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/304\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 33.4715i 1.23968i −0.784727 0.619842i \(-0.787197\pi\)
0.784727 0.619842i \(-0.212803\pi\)
\(4\) 0 0
\(5\) 71.5877 0.572701 0.286351 0.958125i \(-0.407558\pi\)
0.286351 + 0.958125i \(0.407558\pi\)
\(6\) 0 0
\(7\) −51.8109 −0.151052 −0.0755260 0.997144i \(-0.524064\pi\)
−0.0755260 + 0.997144i \(0.524064\pi\)
\(8\) 0 0
\(9\) −391.339 −0.536816
\(10\) 0 0
\(11\) −510.930 −0.383870 −0.191935 0.981408i \(-0.561476\pi\)
−0.191935 + 0.981408i \(0.561476\pi\)
\(12\) 0 0
\(13\) 2026.71i 0.922489i 0.887273 + 0.461244i \(0.152597\pi\)
−0.887273 + 0.461244i \(0.847403\pi\)
\(14\) 0 0
\(15\) 2396.14i 0.709968i
\(16\) 0 0
\(17\) 8792.00 1.78954 0.894769 0.446528i \(-0.147340\pi\)
0.894769 + 0.446528i \(0.147340\pi\)
\(18\) 0 0
\(19\) 6072.98 3188.23i 0.885403 0.464824i
\(20\) 0 0
\(21\) 1734.19i 0.187257i
\(22\) 0 0
\(23\) −22866.7 −1.87940 −0.939702 0.341994i \(-0.888898\pi\)
−0.939702 + 0.341994i \(0.888898\pi\)
\(24\) 0 0
\(25\) −10500.2 −0.672013
\(26\) 0 0
\(27\) 11302.0i 0.574202i
\(28\) 0 0
\(29\) 41913.8i 1.71856i −0.511510 0.859278i \(-0.670914\pi\)
0.511510 0.859278i \(-0.329086\pi\)
\(30\) 0 0
\(31\) 49006.5i 1.64501i −0.568758 0.822505i \(-0.692576\pi\)
0.568758 0.822505i \(-0.307424\pi\)
\(32\) 0 0
\(33\) 17101.6i 0.475877i
\(34\) 0 0
\(35\) −3709.02 −0.0865077
\(36\) 0 0
\(37\) 6756.83i 0.133395i 0.997773 + 0.0666973i \(0.0212462\pi\)
−0.997773 + 0.0666973i \(0.978754\pi\)
\(38\) 0 0
\(39\) 67836.9 1.14359
\(40\) 0 0
\(41\) 39497.7i 0.573087i 0.958067 + 0.286543i \(0.0925063\pi\)
−0.958067 + 0.286543i \(0.907494\pi\)
\(42\) 0 0
\(43\) 36231.6 0.455704 0.227852 0.973696i \(-0.426830\pi\)
0.227852 + 0.973696i \(0.426830\pi\)
\(44\) 0 0
\(45\) −28015.0 −0.307435
\(46\) 0 0
\(47\) 142082. 1.36850 0.684250 0.729247i \(-0.260130\pi\)
0.684250 + 0.729247i \(0.260130\pi\)
\(48\) 0 0
\(49\) −114965. −0.977183
\(50\) 0 0
\(51\) 294281.i 2.21846i
\(52\) 0 0
\(53\) 1332.46i 0.00895009i −0.999990 0.00447504i \(-0.998576\pi\)
0.999990 0.00447504i \(-0.00142446\pi\)
\(54\) 0 0
\(55\) −36576.3 −0.219843
\(56\) 0 0
\(57\) −106715. 203271.i −0.576235 1.09762i
\(58\) 0 0
\(59\) 90431.3i 0.440314i −0.975464 0.220157i \(-0.929343\pi\)
0.975464 0.220157i \(-0.0706570\pi\)
\(60\) 0 0
\(61\) 111587. 0.491613 0.245806 0.969319i \(-0.420947\pi\)
0.245806 + 0.969319i \(0.420947\pi\)
\(62\) 0 0
\(63\) 20275.6 0.0810871
\(64\) 0 0
\(65\) 145087.i 0.528310i
\(66\) 0 0
\(67\) 152853.i 0.508216i 0.967176 + 0.254108i \(0.0817818\pi\)
−0.967176 + 0.254108i \(0.918218\pi\)
\(68\) 0 0
\(69\) 765382.i 2.32987i
\(70\) 0 0
\(71\) 572540.i 1.59967i −0.600219 0.799836i \(-0.704920\pi\)
0.600219 0.799836i \(-0.295080\pi\)
\(72\) 0 0
\(73\) −231113. −0.594094 −0.297047 0.954863i \(-0.596002\pi\)
−0.297047 + 0.954863i \(0.596002\pi\)
\(74\) 0 0
\(75\) 351457.i 0.833084i
\(76\) 0 0
\(77\) 26471.7 0.0579843
\(78\) 0 0
\(79\) 421938.i 0.855790i −0.903828 0.427895i \(-0.859255\pi\)
0.903828 0.427895i \(-0.140745\pi\)
\(80\) 0 0
\(81\) −663581. −1.24864
\(82\) 0 0
\(83\) −767322. −1.34197 −0.670986 0.741470i \(-0.734129\pi\)
−0.670986 + 0.741470i \(0.734129\pi\)
\(84\) 0 0
\(85\) 629399. 1.02487
\(86\) 0 0
\(87\) −1.40292e6 −2.13046
\(88\) 0 0
\(89\) 486637.i 0.690296i −0.938548 0.345148i \(-0.887829\pi\)
0.938548 0.345148i \(-0.112171\pi\)
\(90\) 0 0
\(91\) 105005.i 0.139344i
\(92\) 0 0
\(93\) −1.64032e6 −2.03929
\(94\) 0 0
\(95\) 434750. 228238.i 0.507071 0.266205i
\(96\) 0 0
\(97\) 1.34102e6i 1.46933i 0.678427 + 0.734667i \(0.262662\pi\)
−0.678427 + 0.734667i \(0.737338\pi\)
\(98\) 0 0
\(99\) 199947. 0.206067
\(100\) 0 0
\(101\) −733819. −0.712238 −0.356119 0.934441i \(-0.615900\pi\)
−0.356119 + 0.934441i \(0.615900\pi\)
\(102\) 0 0
\(103\) 536220.i 0.490717i −0.969432 0.245358i \(-0.921094\pi\)
0.969432 0.245358i \(-0.0789057\pi\)
\(104\) 0 0
\(105\) 124146.i 0.107242i
\(106\) 0 0
\(107\) 1.62051e6i 1.32282i 0.750024 + 0.661410i \(0.230042\pi\)
−0.750024 + 0.661410i \(0.769958\pi\)
\(108\) 0 0
\(109\) 944227.i 0.729117i −0.931181 0.364558i \(-0.881220\pi\)
0.931181 0.364558i \(-0.118780\pi\)
\(110\) 0 0
\(111\) 226161. 0.165367
\(112\) 0 0
\(113\) 1.78541e6i 1.23738i −0.785636 0.618690i \(-0.787664\pi\)
0.785636 0.618690i \(-0.212336\pi\)
\(114\) 0 0
\(115\) −1.63697e6 −1.07634
\(116\) 0 0
\(117\) 793129.i 0.495206i
\(118\) 0 0
\(119\) −455521. −0.270314
\(120\) 0 0
\(121\) −1.51051e6 −0.852644
\(122\) 0 0
\(123\) 1.32205e6 0.710446
\(124\) 0 0
\(125\) −1.87024e6 −0.957564
\(126\) 0 0
\(127\) 2.67124e6i 1.30407i 0.758189 + 0.652035i \(0.226085\pi\)
−0.758189 + 0.652035i \(0.773915\pi\)
\(128\) 0 0
\(129\) 1.21273e6i 0.564928i
\(130\) 0 0
\(131\) −4.14444e6 −1.84354 −0.921769 0.387740i \(-0.873256\pi\)
−0.921769 + 0.387740i \(0.873256\pi\)
\(132\) 0 0
\(133\) −314646. + 165185.i −0.133742 + 0.0702127i
\(134\) 0 0
\(135\) 809085.i 0.328846i
\(136\) 0 0
\(137\) 3.08052e6 1.19801 0.599007 0.800744i \(-0.295562\pi\)
0.599007 + 0.800744i \(0.295562\pi\)
\(138\) 0 0
\(139\) −917454. −0.341618 −0.170809 0.985304i \(-0.554638\pi\)
−0.170809 + 0.985304i \(0.554638\pi\)
\(140\) 0 0
\(141\) 4.75569e6i 1.69651i
\(142\) 0 0
\(143\) 1.03551e6i 0.354115i
\(144\) 0 0
\(145\) 3.00051e6i 0.984219i
\(146\) 0 0
\(147\) 3.84803e6i 1.21140i
\(148\) 0 0
\(149\) 2.94518e6 0.890335 0.445168 0.895447i \(-0.353144\pi\)
0.445168 + 0.895447i \(0.353144\pi\)
\(150\) 0 0
\(151\) 4.51351e6i 1.31094i −0.755220 0.655471i \(-0.772470\pi\)
0.755220 0.655471i \(-0.227530\pi\)
\(152\) 0 0
\(153\) −3.44065e6 −0.960652
\(154\) 0 0
\(155\) 3.50826e6i 0.942099i
\(156\) 0 0
\(157\) 3.86325e6 0.998284 0.499142 0.866520i \(-0.333649\pi\)
0.499142 + 0.866520i \(0.333649\pi\)
\(158\) 0 0
\(159\) −44599.5 −0.0110953
\(160\) 0 0
\(161\) 1.18474e6 0.283888
\(162\) 0 0
\(163\) −3.11582e6 −0.719466 −0.359733 0.933055i \(-0.617132\pi\)
−0.359733 + 0.933055i \(0.617132\pi\)
\(164\) 0 0
\(165\) 1.22426e6i 0.272535i
\(166\) 0 0
\(167\) 5.76930e6i 1.23872i −0.785106 0.619361i \(-0.787392\pi\)
0.785106 0.619361i \(-0.212608\pi\)
\(168\) 0 0
\(169\) 719264. 0.149014
\(170\) 0 0
\(171\) −2.37659e6 + 1.24768e6i −0.475298 + 0.249525i
\(172\) 0 0
\(173\) 2.94440e6i 0.568667i −0.958725 0.284333i \(-0.908228\pi\)
0.958725 0.284333i \(-0.0917723\pi\)
\(174\) 0 0
\(175\) 544025. 0.101509
\(176\) 0 0
\(177\) −3.02687e6 −0.545851
\(178\) 0 0
\(179\) 8.24741e6i 1.43800i 0.695011 + 0.718999i \(0.255399\pi\)
−0.695011 + 0.718999i \(0.744601\pi\)
\(180\) 0 0
\(181\) 3.92358e6i 0.661678i −0.943687 0.330839i \(-0.892668\pi\)
0.943687 0.330839i \(-0.107332\pi\)
\(182\) 0 0
\(183\) 3.73497e6i 0.609444i
\(184\) 0 0
\(185\) 483706.i 0.0763952i
\(186\) 0 0
\(187\) −4.49210e6 −0.686950
\(188\) 0 0
\(189\) 585567.i 0.0867344i
\(190\) 0 0
\(191\) −4.67486e6 −0.670917 −0.335458 0.942055i \(-0.608891\pi\)
−0.335458 + 0.942055i \(0.608891\pi\)
\(192\) 0 0
\(193\) 488419.i 0.0679392i 0.999423 + 0.0339696i \(0.0108149\pi\)
−0.999423 + 0.0339696i \(0.989185\pi\)
\(194\) 0 0
\(195\) 4.85628e6 0.654938
\(196\) 0 0
\(197\) −1.02303e7 −1.33810 −0.669049 0.743218i \(-0.733298\pi\)
−0.669049 + 0.743218i \(0.733298\pi\)
\(198\) 0 0
\(199\) 4.76883e6 0.605136 0.302568 0.953128i \(-0.402156\pi\)
0.302568 + 0.953128i \(0.402156\pi\)
\(200\) 0 0
\(201\) 5.11620e6 0.630027
\(202\) 0 0
\(203\) 2.17159e6i 0.259591i
\(204\) 0 0
\(205\) 2.82755e6i 0.328207i
\(206\) 0 0
\(207\) 8.94863e6 1.00889
\(208\) 0 0
\(209\) −3.10287e6 + 1.62896e6i −0.339879 + 0.178432i
\(210\) 0 0
\(211\) 1.81102e6i 0.192786i −0.995343 0.0963929i \(-0.969269\pi\)
0.995343 0.0963929i \(-0.0307305\pi\)
\(212\) 0 0
\(213\) −1.91638e7 −1.98309
\(214\) 0 0
\(215\) 2.59374e6 0.260982
\(216\) 0 0
\(217\) 2.53907e6i 0.248482i
\(218\) 0 0
\(219\) 7.73568e6i 0.736489i
\(220\) 0 0
\(221\) 1.78188e7i 1.65083i
\(222\) 0 0
\(223\) 3.03795e6i 0.273946i −0.990575 0.136973i \(-0.956263\pi\)
0.990575 0.136973i \(-0.0437374\pi\)
\(224\) 0 0
\(225\) 4.10914e6 0.360747
\(226\) 0 0
\(227\) 1.30769e7i 1.11797i −0.829179 0.558983i \(-0.811192\pi\)
0.829179 0.558983i \(-0.188808\pi\)
\(228\) 0 0
\(229\) 8.33095e6 0.693727 0.346863 0.937916i \(-0.387247\pi\)
0.346863 + 0.937916i \(0.387247\pi\)
\(230\) 0 0
\(231\) 886048.i 0.0718822i
\(232\) 0 0
\(233\) 2.12149e7 1.67715 0.838577 0.544783i \(-0.183388\pi\)
0.838577 + 0.544783i \(0.183388\pi\)
\(234\) 0 0
\(235\) 1.01713e7 0.783742
\(236\) 0 0
\(237\) −1.41229e7 −1.06091
\(238\) 0 0
\(239\) 1.03319e7 0.756807 0.378403 0.925641i \(-0.376473\pi\)
0.378403 + 0.925641i \(0.376473\pi\)
\(240\) 0 0
\(241\) 7.06626e6i 0.504823i 0.967620 + 0.252411i \(0.0812236\pi\)
−0.967620 + 0.252411i \(0.918776\pi\)
\(242\) 0 0
\(243\) 1.39719e7i 0.973722i
\(244\) 0 0
\(245\) −8.23005e6 −0.559634
\(246\) 0 0
\(247\) 6.46161e6 + 1.23082e7i 0.428795 + 0.816774i
\(248\) 0 0
\(249\) 2.56834e7i 1.66362i
\(250\) 0 0
\(251\) 5.59405e6 0.353757 0.176878 0.984233i \(-0.443400\pi\)
0.176878 + 0.984233i \(0.443400\pi\)
\(252\) 0 0
\(253\) 1.16833e7 0.721446
\(254\) 0 0
\(255\) 2.10669e7i 1.27052i
\(256\) 0 0
\(257\) 2.74163e7i 1.61514i 0.589775 + 0.807568i \(0.299217\pi\)
−0.589775 + 0.807568i \(0.700783\pi\)
\(258\) 0 0
\(259\) 350077.i 0.0201495i
\(260\) 0 0
\(261\) 1.64025e7i 0.922547i
\(262\) 0 0
\(263\) −1.80126e7 −0.990168 −0.495084 0.868845i \(-0.664863\pi\)
−0.495084 + 0.868845i \(0.664863\pi\)
\(264\) 0 0
\(265\) 95387.8i 0.00512573i
\(266\) 0 0
\(267\) −1.62885e7 −0.855749
\(268\) 0 0
\(269\) 6.68264e6i 0.343314i −0.985157 0.171657i \(-0.945088\pi\)
0.985157 0.171657i \(-0.0549120\pi\)
\(270\) 0 0
\(271\) 4.73961e6 0.238141 0.119071 0.992886i \(-0.462009\pi\)
0.119071 + 0.992886i \(0.462009\pi\)
\(272\) 0 0
\(273\) −3.51469e6 −0.172742
\(274\) 0 0
\(275\) 5.36488e6 0.257965
\(276\) 0 0
\(277\) 4.14107e7 1.94838 0.974190 0.225730i \(-0.0724767\pi\)
0.974190 + 0.225730i \(0.0724767\pi\)
\(278\) 0 0
\(279\) 1.91781e7i 0.883067i
\(280\) 0 0
\(281\) 2.00129e7i 0.901966i −0.892533 0.450983i \(-0.851073\pi\)
0.892533 0.450983i \(-0.148927\pi\)
\(282\) 0 0
\(283\) 7.23503e6 0.319213 0.159607 0.987181i \(-0.448977\pi\)
0.159607 + 0.987181i \(0.448977\pi\)
\(284\) 0 0
\(285\) −7.63945e6 1.45517e7i −0.330010 0.628608i
\(286\) 0 0
\(287\) 2.04641e6i 0.0865660i
\(288\) 0 0
\(289\) 5.31618e7 2.20245
\(290\) 0 0
\(291\) 4.48860e7 1.82151
\(292\) 0 0
\(293\) 6.07240e6i 0.241411i −0.992688 0.120706i \(-0.961484\pi\)
0.992688 0.120706i \(-0.0385157\pi\)
\(294\) 0 0
\(295\) 6.47377e6i 0.252169i
\(296\) 0 0
\(297\) 5.77455e6i 0.220419i
\(298\) 0 0
\(299\) 4.63442e7i 1.73373i
\(300\) 0 0
\(301\) −1.87719e6 −0.0688350
\(302\) 0 0
\(303\) 2.45620e7i 0.882949i
\(304\) 0 0
\(305\) 7.98824e6 0.281547
\(306\) 0 0
\(307\) 1.92735e7i 0.666109i 0.942908 + 0.333054i \(0.108079\pi\)
−0.942908 + 0.333054i \(0.891921\pi\)
\(308\) 0 0
\(309\) −1.79481e7 −0.608334
\(310\) 0 0
\(311\) 1.31848e7 0.438321 0.219160 0.975689i \(-0.429668\pi\)
0.219160 + 0.975689i \(0.429668\pi\)
\(312\) 0 0
\(313\) −4.27464e7 −1.39401 −0.697006 0.717065i \(-0.745485\pi\)
−0.697006 + 0.717065i \(0.745485\pi\)
\(314\) 0 0
\(315\) 1.45148e6 0.0464387
\(316\) 0 0
\(317\) 5.26029e7i 1.65132i −0.564166 0.825662i \(-0.690802\pi\)
0.564166 0.825662i \(-0.309198\pi\)
\(318\) 0 0
\(319\) 2.14151e7i 0.659701i
\(320\) 0 0
\(321\) 5.42409e7 1.63988
\(322\) 0 0
\(323\) 5.33937e7 2.80309e7i 1.58446 0.831821i
\(324\) 0 0
\(325\) 2.12809e7i 0.619925i
\(326\) 0 0
\(327\) −3.16047e7 −0.903874
\(328\) 0 0
\(329\) −7.36138e6 −0.206715
\(330\) 0 0
\(331\) 5.26389e7i 1.45152i 0.687948 + 0.725760i \(0.258512\pi\)
−0.687948 + 0.725760i \(0.741488\pi\)
\(332\) 0 0
\(333\) 2.64421e6i 0.0716083i
\(334\) 0 0
\(335\) 1.09424e7i 0.291056i
\(336\) 0 0
\(337\) 7.10163e7i 1.85553i −0.373162 0.927766i \(-0.621726\pi\)
0.373162 0.927766i \(-0.378274\pi\)
\(338\) 0 0
\(339\) −5.97603e7 −1.53396
\(340\) 0 0
\(341\) 2.50389e7i 0.631469i
\(342\) 0 0
\(343\) 1.20519e7 0.298658
\(344\) 0 0
\(345\) 5.47919e7i 1.33432i
\(346\) 0 0
\(347\) −3.00746e7 −0.719798 −0.359899 0.932991i \(-0.617189\pi\)
−0.359899 + 0.932991i \(0.617189\pi\)
\(348\) 0 0
\(349\) −6.08341e7 −1.43110 −0.715551 0.698560i \(-0.753824\pi\)
−0.715551 + 0.698560i \(0.753824\pi\)
\(350\) 0 0
\(351\) 2.29059e7 0.529695
\(352\) 0 0
\(353\) 3.29214e7 0.748436 0.374218 0.927341i \(-0.377911\pi\)
0.374218 + 0.927341i \(0.377911\pi\)
\(354\) 0 0
\(355\) 4.09868e7i 0.916134i
\(356\) 0 0
\(357\) 1.52470e7i 0.335103i
\(358\) 0 0
\(359\) 7.25010e7 1.56697 0.783485 0.621410i \(-0.213440\pi\)
0.783485 + 0.621410i \(0.213440\pi\)
\(360\) 0 0
\(361\) 2.67163e7 3.87241e7i 0.567877 0.823113i
\(362\) 0 0
\(363\) 5.05590e7i 1.05701i
\(364\) 0 0
\(365\) −1.65448e7 −0.340238
\(366\) 0 0
\(367\) −719204. −0.0145497 −0.00727485 0.999974i \(-0.502316\pi\)
−0.00727485 + 0.999974i \(0.502316\pi\)
\(368\) 0 0
\(369\) 1.54570e7i 0.307642i
\(370\) 0 0
\(371\) 69036.0i 0.00135193i
\(372\) 0 0
\(373\) 8.30946e7i 1.60120i −0.599197 0.800601i \(-0.704514\pi\)
0.599197 0.800601i \(-0.295486\pi\)
\(374\) 0 0
\(375\) 6.25997e7i 1.18708i
\(376\) 0 0
\(377\) 8.49471e7 1.58535
\(378\) 0 0
\(379\) 1.81125e7i 0.332707i −0.986066 0.166353i \(-0.946801\pi\)
0.986066 0.166353i \(-0.0531992\pi\)
\(380\) 0 0
\(381\) 8.94102e7 1.61663
\(382\) 0 0
\(383\) 4.46626e7i 0.794965i 0.917610 + 0.397482i \(0.130116\pi\)
−0.917610 + 0.397482i \(0.869884\pi\)
\(384\) 0 0
\(385\) 1.89505e6 0.0332077
\(386\) 0 0
\(387\) −1.41788e7 −0.244629
\(388\) 0 0
\(389\) 1.94692e7 0.330750 0.165375 0.986231i \(-0.447117\pi\)
0.165375 + 0.986231i \(0.447117\pi\)
\(390\) 0 0
\(391\) −2.01044e8 −3.36327
\(392\) 0 0
\(393\) 1.38720e8i 2.28540i
\(394\) 0 0
\(395\) 3.02055e7i 0.490112i
\(396\) 0 0
\(397\) −7.23981e7 −1.15706 −0.578529 0.815662i \(-0.696373\pi\)
−0.578529 + 0.815662i \(0.696373\pi\)
\(398\) 0 0
\(399\) 5.52898e6 + 1.05317e7i 0.0870415 + 0.165798i
\(400\) 0 0
\(401\) 2.82777e7i 0.438542i −0.975664 0.219271i \(-0.929632\pi\)
0.975664 0.219271i \(-0.0703678\pi\)
\(402\) 0 0
\(403\) 9.93218e7 1.51750
\(404\) 0 0
\(405\) −4.75042e7 −0.715100
\(406\) 0 0
\(407\) 3.45227e6i 0.0512061i
\(408\) 0 0
\(409\) 3.47597e7i 0.508050i −0.967198 0.254025i \(-0.918245\pi\)
0.967198 0.254025i \(-0.0817545\pi\)
\(410\) 0 0
\(411\) 1.03109e8i 1.48516i
\(412\) 0 0
\(413\) 4.68533e6i 0.0665104i
\(414\) 0 0
\(415\) −5.49308e7 −0.768549
\(416\) 0 0
\(417\) 3.07085e7i 0.423498i
\(418\) 0 0
\(419\) −6.00892e7 −0.816872 −0.408436 0.912787i \(-0.633926\pi\)
−0.408436 + 0.912787i \(0.633926\pi\)
\(420\) 0 0
\(421\) 5.86723e7i 0.786297i 0.919475 + 0.393149i \(0.128614\pi\)
−0.919475 + 0.393149i \(0.871386\pi\)
\(422\) 0 0
\(423\) −5.56021e7 −0.734633
\(424\) 0 0
\(425\) −9.23179e7 −1.20259
\(426\) 0 0
\(427\) −5.78141e6 −0.0742591
\(428\) 0 0
\(429\) −3.46599e7 −0.438991
\(430\) 0 0
\(431\) 1.26614e8i 1.58143i 0.612184 + 0.790715i \(0.290291\pi\)
−0.612184 + 0.790715i \(0.709709\pi\)
\(432\) 0 0
\(433\) 6.14600e7i 0.757057i 0.925590 + 0.378529i \(0.123570\pi\)
−0.925590 + 0.378529i \(0.876430\pi\)
\(434\) 0 0
\(435\) −1.00432e8 −1.22012
\(436\) 0 0
\(437\) −1.38869e8 + 7.29043e7i −1.66403 + 0.873593i
\(438\) 0 0
\(439\) 1.93952e7i 0.229246i 0.993409 + 0.114623i \(0.0365660\pi\)
−0.993409 + 0.114623i \(0.963434\pi\)
\(440\) 0 0
\(441\) 4.49901e7 0.524567
\(442\) 0 0
\(443\) 1.20889e8 1.39052 0.695259 0.718759i \(-0.255290\pi\)
0.695259 + 0.718759i \(0.255290\pi\)
\(444\) 0 0
\(445\) 3.48372e7i 0.395333i
\(446\) 0 0
\(447\) 9.85796e7i 1.10373i
\(448\) 0 0
\(449\) 4.67006e7i 0.515921i −0.966155 0.257961i \(-0.916950\pi\)
0.966155 0.257961i \(-0.0830505\pi\)
\(450\) 0 0
\(451\) 2.01806e7i 0.219991i
\(452\) 0 0
\(453\) −1.51074e8 −1.62515
\(454\) 0 0
\(455\) 7.51710e6i 0.0798024i
\(456\) 0 0
\(457\) 1.41054e7 0.147787 0.0738934 0.997266i \(-0.476458\pi\)
0.0738934 + 0.997266i \(0.476458\pi\)
\(458\) 0 0
\(459\) 9.93674e7i 1.02756i
\(460\) 0 0
\(461\) 4.97662e7 0.507962 0.253981 0.967209i \(-0.418260\pi\)
0.253981 + 0.967209i \(0.418260\pi\)
\(462\) 0 0
\(463\) 9.83674e7 0.991079 0.495539 0.868586i \(-0.334970\pi\)
0.495539 + 0.868586i \(0.334970\pi\)
\(464\) 0 0
\(465\) −1.17427e8 −1.16790
\(466\) 0 0
\(467\) 6.59046e7 0.647090 0.323545 0.946213i \(-0.395125\pi\)
0.323545 + 0.946213i \(0.395125\pi\)
\(468\) 0 0
\(469\) 7.91942e6i 0.0767671i
\(470\) 0 0
\(471\) 1.29309e8i 1.23756i
\(472\) 0 0
\(473\) −1.85118e7 −0.174931
\(474\) 0 0
\(475\) −6.37675e7 + 3.34771e7i −0.595003 + 0.312368i
\(476\) 0 0
\(477\) 521444.i 0.00480455i
\(478\) 0 0
\(479\) −4.08450e7 −0.371649 −0.185824 0.982583i \(-0.559496\pi\)
−0.185824 + 0.982583i \(0.559496\pi\)
\(480\) 0 0
\(481\) −1.36941e7 −0.123055
\(482\) 0 0
\(483\) 3.96551e7i 0.351931i
\(484\) 0 0
\(485\) 9.60006e7i 0.841490i
\(486\) 0 0
\(487\) 4.20128e7i 0.363743i −0.983322 0.181871i \(-0.941785\pi\)
0.983322 0.181871i \(-0.0582155\pi\)
\(488\) 0 0
\(489\) 1.04291e8i 0.891910i
\(490\) 0 0
\(491\) −1.24183e8 −1.04910 −0.524549 0.851380i \(-0.675766\pi\)
−0.524549 + 0.851380i \(0.675766\pi\)
\(492\) 0 0
\(493\) 3.68507e8i 3.07542i
\(494\) 0 0
\(495\) 1.43137e7 0.118015
\(496\) 0 0
\(497\) 2.96638e7i 0.241634i
\(498\) 0 0
\(499\) 1.52533e7 0.122762 0.0613809 0.998114i \(-0.480450\pi\)
0.0613809 + 0.998114i \(0.480450\pi\)
\(500\) 0 0
\(501\) −1.93107e8 −1.53562
\(502\) 0 0
\(503\) 1.71504e8 1.34763 0.673814 0.738901i \(-0.264655\pi\)
0.673814 + 0.738901i \(0.264655\pi\)
\(504\) 0 0
\(505\) −5.25324e7 −0.407899
\(506\) 0 0
\(507\) 2.40748e7i 0.184731i
\(508\) 0 0
\(509\) 7.95755e7i 0.603429i −0.953398 0.301714i \(-0.902441\pi\)
0.953398 0.301714i \(-0.0975589\pi\)
\(510\) 0 0
\(511\) 1.19742e7 0.0897392
\(512\) 0 0
\(513\) −3.60334e7 6.86369e7i −0.266903 0.508400i
\(514\) 0 0
\(515\) 3.83867e7i 0.281034i
\(516\) 0 0
\(517\) −7.25939e7 −0.525326
\(518\) 0 0
\(519\) −9.85532e7 −0.704967
\(520\) 0 0
\(521\) 2.40151e8i 1.69813i 0.528288 + 0.849065i \(0.322834\pi\)
−0.528288 + 0.849065i \(0.677166\pi\)
\(522\) 0 0
\(523\) 8.35375e7i 0.583951i 0.956426 + 0.291975i \(0.0943125\pi\)
−0.956426 + 0.291975i \(0.905687\pi\)
\(524\) 0 0
\(525\) 1.82093e7i 0.125839i
\(526\) 0 0
\(527\) 4.30865e8i 2.94381i
\(528\) 0 0
\(529\) 3.74851e8 2.53216
\(530\) 0 0
\(531\) 3.53893e7i 0.236368i
\(532\) 0 0
\(533\) −8.00503e7 −0.528666
\(534\) 0 0
\(535\) 1.16009e8i 0.757581i
\(536\) 0 0
\(537\) 2.76053e8 1.78266
\(538\) 0 0
\(539\) 5.87389e7 0.375111
\(540\) 0 0
\(541\) −1.33819e8 −0.845133 −0.422567 0.906332i \(-0.638871\pi\)
−0.422567 + 0.906332i \(0.638871\pi\)
\(542\) 0 0
\(543\) −1.31328e8 −0.820271
\(544\) 0 0
\(545\) 6.75950e7i 0.417566i
\(546\) 0 0
\(547\) 1.97530e7i 0.120690i 0.998178 + 0.0603449i \(0.0192200\pi\)
−0.998178 + 0.0603449i \(0.980780\pi\)
\(548\) 0 0
\(549\) −4.36682e7 −0.263905
\(550\) 0 0
\(551\) −1.33631e8 2.54542e8i −0.798826 1.52161i
\(552\) 0 0
\(553\) 2.18610e7i 0.129269i
\(554\) 0 0
\(555\) 1.61903e7 0.0947059
\(556\) 0 0
\(557\) 1.64574e7 0.0952349 0.0476175 0.998866i \(-0.484837\pi\)
0.0476175 + 0.998866i \(0.484837\pi\)
\(558\) 0 0
\(559\) 7.34309e7i 0.420382i
\(560\) 0 0
\(561\) 1.50357e8i 0.851600i
\(562\) 0 0
\(563\) 2.80150e6i 0.0156988i 0.999969 + 0.00784939i \(0.00249856\pi\)
−0.999969 + 0.00784939i \(0.997501\pi\)
\(564\) 0 0
\(565\) 1.27813e8i 0.708648i
\(566\) 0 0
\(567\) 3.43807e7 0.188610
\(568\) 0 0
\(569\) 1.04615e8i 0.567881i −0.958842 0.283941i \(-0.908358\pi\)
0.958842 0.283941i \(-0.0916418\pi\)
\(570\) 0 0
\(571\) −2.28204e8 −1.22579 −0.612893 0.790166i \(-0.709994\pi\)
−0.612893 + 0.790166i \(0.709994\pi\)
\(572\) 0 0
\(573\) 1.56474e8i 0.831725i
\(574\) 0 0
\(575\) 2.40105e8 1.26298
\(576\) 0 0
\(577\) 2.96754e8 1.54479 0.772395 0.635142i \(-0.219058\pi\)
0.772395 + 0.635142i \(0.219058\pi\)
\(578\) 0 0
\(579\) 1.63481e7 0.0842231
\(580\) 0 0
\(581\) 3.97556e7 0.202708
\(582\) 0 0
\(583\) 680795.i 0.00343567i
\(584\) 0 0
\(585\) 5.67783e7i 0.283605i
\(586\) 0 0
\(587\) 1.18280e7 0.0584785 0.0292392 0.999572i \(-0.490692\pi\)
0.0292392 + 0.999572i \(0.490692\pi\)
\(588\) 0 0
\(589\) −1.56244e8 2.97615e8i −0.764640 1.45650i
\(590\) 0 0
\(591\) 3.42422e8i 1.65882i
\(592\) 0 0
\(593\) 1.29365e8 0.620374 0.310187 0.950676i \(-0.399608\pi\)
0.310187 + 0.950676i \(0.399608\pi\)
\(594\) 0 0
\(595\) −3.26097e7 −0.154809
\(596\) 0 0
\(597\) 1.59620e8i 0.750177i
\(598\) 0 0
\(599\) 2.75609e7i 0.128237i 0.997942 + 0.0641184i \(0.0204235\pi\)
−0.997942 + 0.0641184i \(0.979576\pi\)
\(600\) 0 0
\(601\) 2.62146e7i 0.120759i 0.998175 + 0.0603796i \(0.0192311\pi\)
−0.998175 + 0.0603796i \(0.980769\pi\)
\(602\) 0 0
\(603\) 5.98171e7i 0.272818i
\(604\) 0 0
\(605\) −1.08134e8 −0.488310
\(606\) 0 0
\(607\) 1.81199e7i 0.0810196i −0.999179 0.0405098i \(-0.987102\pi\)
0.999179 0.0405098i \(-0.0128982\pi\)
\(608\) 0 0
\(609\) 7.26864e7 0.321811
\(610\) 0 0
\(611\) 2.87958e8i 1.26243i
\(612\) 0 0
\(613\) −1.45891e8 −0.633354 −0.316677 0.948533i \(-0.602567\pi\)
−0.316677 + 0.948533i \(0.602567\pi\)
\(614\) 0 0
\(615\) 9.46422e7 0.406873
\(616\) 0 0
\(617\) 2.65303e8 1.12950 0.564752 0.825261i \(-0.308972\pi\)
0.564752 + 0.825261i \(0.308972\pi\)
\(618\) 0 0
\(619\) −4.21064e7 −0.177532 −0.0887658 0.996053i \(-0.528292\pi\)
−0.0887658 + 0.996053i \(0.528292\pi\)
\(620\) 0 0
\(621\) 2.58440e8i 1.07916i
\(622\) 0 0
\(623\) 2.52131e7i 0.104271i
\(624\) 0 0
\(625\) 3.01795e7 0.123615
\(626\) 0 0
\(627\) 5.45238e7 + 1.03858e8i 0.221199 + 0.421343i
\(628\) 0 0
\(629\) 5.94061e7i 0.238715i
\(630\) 0 0
\(631\) −2.37249e7 −0.0944314 −0.0472157 0.998885i \(-0.515035\pi\)
−0.0472157 + 0.998885i \(0.515035\pi\)
\(632\) 0 0
\(633\) −6.06173e7 −0.238993
\(634\) 0 0
\(635\) 1.91227e8i 0.746843i
\(636\) 0 0
\(637\) 2.33000e8i 0.901441i
\(638\) 0 0
\(639\) 2.24057e8i 0.858729i
\(640\) 0 0
\(641\) 7.10625e7i 0.269815i 0.990858 + 0.134908i \(0.0430737\pi\)
−0.990858 + 0.134908i \(0.956926\pi\)
\(642\) 0 0
\(643\) −5.19477e7 −0.195404 −0.0977021 0.995216i \(-0.531149\pi\)
−0.0977021 + 0.995216i \(0.531149\pi\)
\(644\) 0 0
\(645\) 8.68162e7i 0.323535i
\(646\) 0 0
\(647\) 1.98051e8 0.731246 0.365623 0.930763i \(-0.380856\pi\)
0.365623 + 0.930763i \(0.380856\pi\)
\(648\) 0 0
\(649\) 4.62041e7i 0.169023i
\(650\) 0 0
\(651\) 8.49863e7 0.308039
\(652\) 0 0
\(653\) 1.19882e8 0.430540 0.215270 0.976555i \(-0.430937\pi\)
0.215270 + 0.976555i \(0.430937\pi\)
\(654\) 0 0
\(655\) −2.96691e8 −1.05580
\(656\) 0 0
\(657\) 9.04433e7 0.318919
\(658\) 0 0
\(659\) 4.57886e8i 1.59993i 0.600045 + 0.799966i \(0.295149\pi\)
−0.600045 + 0.799966i \(0.704851\pi\)
\(660\) 0 0
\(661\) 3.03652e8i 1.05141i 0.850667 + 0.525705i \(0.176198\pi\)
−0.850667 + 0.525705i \(0.823802\pi\)
\(662\) 0 0
\(663\) 5.96422e8 2.04651
\(664\) 0 0
\(665\) −2.25248e7 + 1.18252e7i −0.0765942 + 0.0402109i
\(666\) 0 0
\(667\) 9.58432e8i 3.22986i
\(668\) 0 0
\(669\) −1.01684e8 −0.339607
\(670\) 0 0
\(671\) −5.70131e7 −0.188715
\(672\) 0 0
\(673\) 4.98432e8i 1.63516i −0.575813 0.817582i \(-0.695314\pi\)
0.575813 0.817582i \(-0.304686\pi\)
\(674\) 0 0
\(675\) 1.18674e8i 0.385871i
\(676\) 0 0
\(677\) 1.01610e8i 0.327470i 0.986504 + 0.163735i \(0.0523542\pi\)
−0.986504 + 0.163735i \(0.947646\pi\)
\(678\) 0 0
\(679\) 6.94795e7i 0.221946i
\(680\) 0 0
\(681\) −4.37705e8 −1.38593
\(682\) 0 0
\(683\) 2.15235e8i 0.675539i −0.941229 0.337770i \(-0.890328\pi\)
0.941229 0.337770i \(-0.109672\pi\)
\(684\) 0 0
\(685\) 2.20527e8 0.686104
\(686\) 0 0
\(687\) 2.78849e8i 0.860001i
\(688\) 0 0
\(689\) 2.70051e6 0.00825636
\(690\) 0 0
\(691\) −4.21330e8 −1.27699 −0.638497 0.769625i \(-0.720443\pi\)
−0.638497 + 0.769625i \(0.720443\pi\)
\(692\) 0 0
\(693\) −1.03594e7 −0.0311269
\(694\) 0 0
\(695\) −6.56784e7 −0.195645
\(696\) 0 0
\(697\) 3.47264e8i 1.02556i
\(698\) 0 0
\(699\) 7.10093e8i 2.07914i
\(700\) 0 0
\(701\) 6.22655e8 1.80756 0.903782 0.427994i \(-0.140780\pi\)
0.903782 + 0.427994i \(0.140780\pi\)
\(702\) 0 0
\(703\) 2.15423e7 + 4.10341e7i 0.0620050 + 0.118108i
\(704\) 0 0
\(705\) 3.40448e8i 0.971592i
\(706\) 0 0
\(707\) 3.80198e7 0.107585
\(708\) 0 0
\(709\) 1.65405e8 0.464097 0.232049 0.972704i \(-0.425457\pi\)
0.232049 + 0.972704i \(0.425457\pi\)
\(710\) 0 0
\(711\) 1.65121e8i 0.459402i
\(712\) 0 0
\(713\) 1.12062e9i 3.09164i
\(714\) 0 0
\(715\) 7.41295e7i 0.202802i
\(716\) 0 0
\(717\) 3.45823e8i 0.938201i
\(718\) 0 0
\(719\) 3.39243e8 0.912692 0.456346 0.889802i \(-0.349158\pi\)
0.456346 + 0.889802i \(0.349158\pi\)
\(720\) 0 0
\(721\) 2.77820e7i 0.0741238i
\(722\) 0 0
\(723\) 2.36518e8 0.625820
\(724\) 0 0
\(725\) 4.40104e8i 1.15489i
\(726\) 0 0
\(727\) −3.80897e8 −0.991297 −0.495649 0.868523i \(-0.665070\pi\)
−0.495649 + 0.868523i \(0.665070\pi\)
\(728\) 0 0
\(729\) −1.60923e7 −0.0415369
\(730\) 0 0
\(731\) 3.18549e8 0.815499
\(732\) 0 0
\(733\) −2.61548e8 −0.664108 −0.332054 0.943260i \(-0.607742\pi\)
−0.332054 + 0.943260i \(0.607742\pi\)
\(734\) 0 0
\(735\) 2.75472e8i 0.693769i
\(736\) 0 0
\(737\) 7.80970e7i 0.195089i
\(738\) 0 0
\(739\) −4.88512e8 −1.21044 −0.605218 0.796060i \(-0.706914\pi\)
−0.605218 + 0.796060i \(0.706914\pi\)
\(740\) 0 0
\(741\) 4.11972e8 2.16279e8i 1.01254 0.531570i
\(742\) 0 0
\(743\) 3.47484e8i 0.847166i −0.905857 0.423583i \(-0.860772\pi\)
0.905857 0.423583i \(-0.139228\pi\)
\(744\) 0 0
\(745\) 2.10839e8 0.509896
\(746\) 0 0
\(747\) 3.00283e8 0.720392
\(748\) 0 0
\(749\) 8.39602e7i 0.199815i
\(750\) 0 0
\(751\) 5.37037e8i 1.26790i −0.773375 0.633949i \(-0.781433\pi\)
0.773375 0.633949i \(-0.218567\pi\)
\(752\) 0 0
\(753\) 1.87241e8i 0.438547i
\(754\) 0 0
\(755\) 3.23111e8i 0.750778i
\(756\) 0 0
\(757\) −4.06677e7 −0.0937479 −0.0468740 0.998901i \(-0.514926\pi\)
−0.0468740 + 0.998901i \(0.514926\pi\)
\(758\) 0 0
\(759\) 3.91057e8i 0.894365i
\(760\) 0 0
\(761\) 3.59362e7 0.0815413 0.0407707 0.999169i \(-0.487019\pi\)
0.0407707 + 0.999169i \(0.487019\pi\)
\(762\) 0 0
\(763\) 4.89212e7i 0.110135i
\(764\) 0 0
\(765\) −2.46308e8 −0.550167
\(766\) 0 0
\(767\) 1.83278e8 0.406185
\(768\) 0 0
\(769\) 5.90636e8 1.29880 0.649398 0.760449i \(-0.275021\pi\)
0.649398 + 0.760449i \(0.275021\pi\)
\(770\) 0 0
\(771\) 9.17663e8 2.00226
\(772\) 0 0
\(773\) 5.37852e8i 1.16446i 0.813025 + 0.582229i \(0.197819\pi\)
−0.813025 + 0.582229i \(0.802181\pi\)
\(774\) 0 0
\(775\) 5.14578e8i 1.10547i
\(776\) 0 0
\(777\) −1.17176e7 −0.0249790
\(778\) 0 0
\(779\) 1.25928e8 + 2.39869e8i 0.266385 + 0.507413i
\(780\) 0 0
\(781\) 2.92528e8i 0.614065i
\(782\) 0 0
\(783\) −4.73711e8 −0.986798
\(784\) 0 0
\(785\) 2.76561e8 0.571719
\(786\) 0 0
\(787\) 4.32782e8i 0.887860i −0.896061 0.443930i \(-0.853584\pi\)
0.896061 0.443930i \(-0.146416\pi\)
\(788\) 0 0
\(789\) 6.02908e8i 1.22750i
\(790\) 0 0
\(791\) 9.25037e7i 0.186909i
\(792\) 0 0
\(793\) 2.26154e8i 0.453507i
\(794\) 0 0
\(795\) −3.19277e6 −0.00635428
\(796\) 0 0
\(797\) 7.42582e8i 1.46680i −0.679800 0.733398i \(-0.737933\pi\)
0.679800 0.733398i \(-0.262067\pi\)
\(798\) 0 0
\(799\) 1.24918e9 2.44898
\(800\) 0 0
\(801\) 1.90440e8i 0.370562i
\(802\) 0 0
\(803\) 1.18083e8 0.228055
\(804\) 0 0
\(805\) 8.48131e7 0.162583
\(806\) 0 0
\(807\) −2.23678e8 −0.425600
\(808\) 0 0
\(809\) 2.67711e8 0.505616 0.252808 0.967516i \(-0.418646\pi\)
0.252808 + 0.967516i \(0.418646\pi\)
\(810\) 0 0
\(811\) 6.36834e8i 1.19389i 0.802283 + 0.596944i \(0.203619\pi\)
−0.802283 + 0.596944i \(0.796381\pi\)
\(812\) 0 0
\(813\) 1.58642e8i 0.295220i
\(814\) 0 0
\(815\) −2.23055e8 −0.412039
\(816\) 0 0
\(817\) 2.20034e8 1.15515e8i 0.403481 0.211822i
\(818\) 0 0
\(819\) 4.10927e7i 0.0748020i
\(820\) 0 0
\(821\) 3.37818e7 0.0610454 0.0305227 0.999534i \(-0.490283\pi\)
0.0305227 + 0.999534i \(0.490283\pi\)
\(822\) 0 0
\(823\) 5.68249e8 1.01939 0.509693 0.860356i \(-0.329759\pi\)
0.509693 + 0.860356i \(0.329759\pi\)
\(824\) 0 0
\(825\) 1.79570e8i 0.319796i
\(826\) 0 0
\(827\) 6.00761e8i 1.06215i −0.847325 0.531074i \(-0.821788\pi\)
0.847325 0.531074i \(-0.178212\pi\)
\(828\) 0 0
\(829\) 5.05498e8i 0.887270i −0.896208 0.443635i \(-0.853689\pi\)
0.896208 0.443635i \(-0.146311\pi\)
\(830\) 0 0
\(831\) 1.38608e9i 2.41537i
\(832\) 0 0
\(833\) −1.01077e9 −1.74871
\(834\) 0 0
\(835\) 4.13011e8i 0.709418i
\(836\) 0 0
\(837\) −5.53872e8 −0.944568
\(838\) 0 0
\(839\) 4.06603e7i 0.0688469i 0.999407 + 0.0344235i \(0.0109595\pi\)
−0.999407 + 0.0344235i \(0.989041\pi\)
\(840\) 0 0
\(841\) −1.16195e9 −1.95343
\(842\) 0 0
\(843\) −6.69860e8 −1.11815
\(844\) 0 0
\(845\) 5.14904e7 0.0853407
\(846\) 0 0
\(847\) 7.82609e7 0.128794
\(848\) 0 0
\(849\) 2.42167e8i 0.395724i
\(850\) 0 0
\(851\) 1.54507e8i 0.250702i
\(852\) 0 0
\(853\) −6.06743e8 −0.977592 −0.488796 0.872398i \(-0.662564\pi\)
−0.488796 + 0.872398i \(0.662564\pi\)
\(854\) 0 0
\(855\) −1.70135e8 + 8.93183e7i −0.272204 + 0.142903i
\(856\) 0 0
\(857\) 1.60934e8i 0.255685i 0.991794 + 0.127843i \(0.0408053\pi\)
−0.991794 + 0.127843i \(0.959195\pi\)
\(858\) 0 0
\(859\) 9.17471e8 1.44748 0.723740 0.690073i \(-0.242421\pi\)
0.723740 + 0.690073i \(0.242421\pi\)
\(860\) 0 0
\(861\) −6.84964e7 −0.107314
\(862\) 0 0
\(863\) 1.03177e9i 1.60528i 0.596466 + 0.802638i \(0.296571\pi\)
−0.596466 + 0.802638i \(0.703429\pi\)
\(864\) 0 0
\(865\) 2.10782e8i 0.325676i
\(866\) 0 0
\(867\) 1.77940e9i 2.73034i
\(868\) 0 0
\(869\) 2.15581e8i 0.328512i
\(870\) 0 0
\(871\) −3.09787e8 −0.468823
\(872\) 0 0
\(873\) 5.24794e8i 0.788762i
\(874\) 0 0
\(875\) 9.68989e7 0.144642
\(876\) 0 0
\(877\) 1.27402e9i 1.88876i 0.328850 + 0.944382i \(0.393339\pi\)
−0.328850 + 0.944382i \(0.606661\pi\)
\(878\) 0 0
\(879\) −2.03252e8 −0.299274
\(880\) 0 0
\(881\) −1.06039e8 −0.155074 −0.0775370 0.996989i \(-0.524706\pi\)
−0.0775370 + 0.996989i \(0.524706\pi\)
\(882\) 0 0
\(883\) −6.63067e8 −0.963108 −0.481554 0.876416i \(-0.659927\pi\)
−0.481554 + 0.876416i \(0.659927\pi\)
\(884\) 0 0
\(885\) −2.16686e8 −0.312609
\(886\) 0 0
\(887\) 6.90226e8i 0.989056i 0.869162 + 0.494528i \(0.164659\pi\)
−0.869162 + 0.494528i \(0.835341\pi\)
\(888\) 0 0
\(889\) 1.38399e8i 0.196983i
\(890\) 0 0
\(891\) 3.39044e8 0.479317
\(892\) 0 0
\(893\) 8.62860e8 4.52989e8i 1.21167 0.636112i
\(894\) 0 0
\(895\) 5.90412e8i 0.823543i
\(896\) 0 0
\(897\) −1.55121e9 −2.14928
\(898\) 0 0
\(899\) −2.05405e9 −2.82704
\(900\) 0 0
\(901\) 1.17150e7i 0.0160165i
\(902\) 0 0
\(903\) 6.28324e7i 0.0853336i
\(904\) 0 0
\(905\) 2.80880e8i 0.378944i
\(906\) 0 0
\(907\) 7.18182e8i 0.962527i −0.876576 0.481263i \(-0.840178\pi\)
0.876576 0.481263i \(-0.159822\pi\)
\(908\) 0 0
\(909\) 2.87172e8 0.382340
\(910\) 0 0
\(911\) 1.37912e9i 1.82410i 0.410083 + 0.912048i \(0.365500\pi\)
−0.410083 + 0.912048i \(0.634500\pi\)
\(912\) 0 0
\(913\) 3.92048e8 0.515142
\(914\) 0 0
\(915\) 2.67378e8i 0.349030i
\(916\) 0 0
\(917\) 2.14727e8 0.278470
\(918\) 0 0
\(919\) −1.20524e9 −1.55284 −0.776419 0.630217i \(-0.782966\pi\)
−0.776419 + 0.630217i \(0.782966\pi\)
\(920\) 0 0
\(921\) 6.45111e8 0.825764
\(922\) 0 0
\(923\) 1.16037e9 1.47568
\(924\) 0 0
\(925\) 7.09482e7i 0.0896429i
\(926\) 0 0
\(927\) 2.09843e8i 0.263425i
\(928\) 0 0
\(929\) −4.36214e7 −0.0544067 −0.0272033 0.999630i \(-0.508660\pi\)
−0.0272033 + 0.999630i \(0.508660\pi\)
\(930\) 0 0
\(931\) −6.98178e8 + 3.66534e8i −0.865201 + 0.454218i
\(932\) 0 0
\(933\) 4.41314e8i 0.543379i
\(934\) 0 0
\(935\) −3.21579e8 −0.393417
\(936\) 0 0
\(937\) 5.87075e8 0.713633 0.356817 0.934174i \(-0.383862\pi\)
0.356817 + 0.934174i \(0.383862\pi\)
\(938\) 0 0
\(939\) 1.43078e9i 1.72813i
\(940\) 0 0
\(941\) 6.17508e8i 0.741094i 0.928814 + 0.370547i \(0.120830\pi\)
−0.928814 + 0.370547i \(0.879170\pi\)
\(942\) 0 0
\(943\) 9.03183e8i 1.07706i
\(944\) 0 0
\(945\) 4.19194e7i 0.0496729i
\(946\) 0 0
\(947\) 1.09790e9 1.29275 0.646375 0.763020i \(-0.276284\pi\)
0.646375 + 0.763020i \(0.276284\pi\)
\(948\) 0 0
\(949\) 4.68398e8i 0.548045i
\(950\) 0 0
\(951\) −1.76070e9 −2.04712
\(952\) 0 0
\(953\) 1.46976e9i 1.69812i 0.528299 + 0.849058i \(0.322830\pi\)
−0.528299 + 0.849058i \(0.677170\pi\)
\(954\) 0 0
\(955\) −3.34662e8 −0.384235
\(956\) 0 0
\(957\) 7.16793e8 0.817821
\(958\) 0 0
\(959\) −1.59604e8 −0.180962
\(960\) 0 0
\(961\) −1.51413e9 −1.70606
\(962\) 0 0
\(963\) 6.34169e8i 0.710111i
\(964\) 0 0
\(965\) 3.49647e7i 0.0389089i
\(966\) 0 0
\(967\) −1.38188e9 −1.52824 −0.764121 0.645072i \(-0.776827\pi\)
−0.764121 + 0.645072i \(0.776827\pi\)
\(968\) 0 0
\(969\) −9.38236e8 1.78716e9i −1.03119 1.96423i
\(970\) 0 0
\(971\) 1.08548e9i 1.18567i 0.805326 + 0.592833i \(0.201991\pi\)
−0.805326 + 0.592833i \(0.798009\pi\)
\(972\) 0 0
\(973\) 4.75341e7 0.0516020
\(974\) 0 0
\(975\) −7.12301e8 −0.768511
\(976\) 0 0
\(977\) 8.78848e8i 0.942388i −0.882029 0.471194i \(-0.843823\pi\)
0.882029 0.471194i \(-0.156177\pi\)
\(978\) 0 0
\(979\) 2.48638e8i 0.264984i
\(980\) 0 0
\(981\) 3.69513e8i 0.391401i
\(982\) 0 0
\(983\) 6.03866e8i 0.635740i −0.948134 0.317870i \(-0.897032\pi\)
0.948134 0.317870i \(-0.102968\pi\)
\(984\) 0 0
\(985\) −7.32360e8 −0.766330
\(986\) 0 0
\(987\) 2.46396e8i 0.256261i
\(988\) 0 0
\(989\) −8.28498e8 −0.856452
\(990\) 0 0
\(991\) 4.53282e8i 0.465745i 0.972507 + 0.232872i \(0.0748124\pi\)
−0.972507 + 0.232872i \(0.925188\pi\)
\(992\) 0 0
\(993\) 1.76190e9 1.79942
\(994\) 0 0
\(995\) 3.41389e8 0.346562
\(996\) 0 0
\(997\) −1.50572e8 −0.151935 −0.0759675 0.997110i \(-0.524205\pi\)
−0.0759675 + 0.997110i \(0.524205\pi\)
\(998\) 0 0
\(999\) 7.63659e7 0.0765954
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 304.7.e.f.113.6 30
4.3 odd 2 152.7.e.a.113.25 yes 30
19.18 odd 2 inner 304.7.e.f.113.25 30
76.75 even 2 152.7.e.a.113.6 30
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.7.e.a.113.6 30 76.75 even 2
152.7.e.a.113.25 yes 30 4.3 odd 2
304.7.e.f.113.6 30 1.1 even 1 trivial
304.7.e.f.113.25 30 19.18 odd 2 inner